Theorem iunsuc 4398

Description: Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Hypotheses

Ref	Expression
iunsuc.1	|- A e. _V
iunsuc.2	|- ( x = A -> B = C )

Assertion

Ref	Expression
iunsuc	|- U_ x e. suc A B = ( U_ x e. A B u. C )

Distinct variable groups:   x, A   x, C

Allowed substitution hint:   B( x)

Proof of Theorem iunsuc

Step	Hyp	Ref	Expression
1		df-suc 4349	. . 3 |- suc A = ( A u. { A } )
2		iuneq1 3879	. . 3 |- ( suc A = ( A u. { A } ) -> U_ x e. suc A B = U_ x e. ( A u. { A } ) B )
3	1, 2	ax-mp 5	. 2 |- U_ x e. suc A B = U_ x e. ( A u. { A } ) B
4		iunxun 3945	. 2 |- U_ x e. ( A u. { A } ) B = ( U_ x e. A B u. U_ x e. { A } B )
5		iunsuc.1	. . . 4 |- A e. _V
6		iunsuc.2	. . . 4 |- ( x = A -> B = C )
7	5, 6	iunxsn 3942	. . 3 |- U_ x e. { A } B = C
8	7	uneq2i 3273	. 2 |- ( U_ x e. A B u. U_ x e. { A } B ) = ( U_ x e. A B u. C )
9	3, 4, 8	3eqtri 2190	1 |- U_ x e. suc A B = ( U_ x e. A B u. C )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   _Vcvv 2726   u. cun 3114   {csn 3576   U_ciun 3866   suc csuc 4343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-iun 3868  df-suc 4349

This theorem is referenced by: (None)